POWER SYSTEM PARAMETER ESTIMATION JOÃO B. A. LONDON JR. AND NEWTON G. BRETAS * LACO, Dep. de Eng. Elétrica, Escola de Engenharia de São Carlos - Universidade de São Paulo Av. TrabalhadorSão Carlense n° 400, CEP: 13560-250, fone: 55 – 16 - 33739321 São Carlos, São Paulo, Brasil E-mails:
[email protected],
[email protected] Abstract This paper presents an approach to parameter/state estimation based on Normal Equations. The proposed approach increases both the state vector, with the parameters to be estimated, and the measurement vector to consider all the measurements that vary over a small range in a certain period of time. Consequently, several scans of measurements are treated as one unique scan and the parameters and states are estimated just once. One new method to observability analysis to this augmented model is also proposed. To verify the performance of the proposed approach, small-scale system examples are presented. Keywords State Estimation, Parameter Estimation, Normal Equations, Observability analysis. Resumo Neste trabalho propõe-se um estimador de estados e parâmetros, baseado nas equações normais. O estimador proposto aumenta tanto o vetor de estados quanto o vetor de medidas. O aumento do vetor de estados será realizado para a inclusão dos parâmetros a serem estimados; realizar-se-á o aumento do vetor de medidas, para considerar medidas de diversas amostras, desde que não tenham sofrido uma alteração significativa dos seus valores. Propõe-se também um método para análise de observabilidade para o modelo aumentado. Através de três exemplos, mostra-se a viabilidade dos métodos propostos. Palavras-chave Estimação de Estados, Estimação de Parâmetros, Equações Normais, Análise de Observabilidade.
1
Introduction
The state estimator is an essential tool to guarantee a secure power system real-time operation. The state estimation process consists in determining the power systems states (voltages and phase angles) through two kinds of information: real-time measurements (analog measurements and topological information, that is, the status of switching devices) as well as static network data (power system parameters). The determination of the true states depends on the reliability of the available information, since errors in the information can lead the state estimation process to wrong results. In a general way, the state estimation process is subject to three types of errors: (i)errors on analogical measurements (gross errors); (ii)incorrect topological information (topological errors) and (iii)errors on the static network data (parameter errors). Parameters errors can have adverse impact on the state estimation process and are less evident than gross and topological errors. Comparatively, the number of papers addressing parameter errors is modest in relation to that devoted to gross and topological errors [Zarco and Expósito, 2000; Abur and Expósito, 2004]. Considering the existing methods to treat power system parameter errors, it can be stated that those which increase the state vector in order to consider the parameters have obtained better results than those based on residual sensitivity analysis [Liu and Lun, 1992; Van Cutsen and Quintana, 1988]. According to the treatment given to the augmented model, the methods that increase the state vector can be divided
in two groups: (i) methods based on Normal Equations [Alsaç et al, 1998; Liu and Lun, 1995]; (ii) methods based on Kalman Filter [Debs, 1974; Slutsker and Clements, 1996]. The methods of the first group have observability problems, since there are rarely enough measurements to enable the estimation of the augmented state vector. To overcome such limitation, the methods of the second group increase the measurement vector considering as pseudomeasurements the previously augmented estimated state. However, the problem of these methods is defining a transition matrix relating the augmented states at consecutive scan times. Considering the assumption that a subset of the measurements varies over a small range in a certain time period, in this paper an approach to parameter/state estimation is proposed. This approach is based on Normal Equations, but it diminishes the possibility of observability problem, since the increase of the state vector will be accompanied by the increase of the measurement vector. The measurement vector will consider all the measurements that vary over a small range in a certain period of time (measurements of several snapshots). As these measurements remain nearly constant over this time period, one can assume that the respective states are also nearly constant in that same period of time. The parameters to be estimated by the proposed method are the series conductances and susceptances of the transmission lines as well as their shunt susceptances. Basing on the method to observability analysis developed by Bretas [Bretas, 1996], an appropriate method to observability analysis is also proposed. This paper is organized as follows: initially the formulation of the power system state estimation
using normal equations is revised (section 2); in section 3 a review of the method proposed by Bretas [Bretas, 1996] to observability analysis is presented; the proposed approach is developed in section 4; in section 5 some examples are presented while the conclusions are in section 6. 2 State Estimation using Normal Equations Power system state estimation is closely related to the statistic regression methods. The non-linear equations relating the (mx1) measurement vector “z” and the (nx1) state vector “x” are: z = h( x ) + w (1) where “w” is an (mx1) random noise vector with zero mean jointly Gaussian distribution and h(.) is a vector-valued non-linear functions that relates the measurements to the system states. Through the conventional Weighted Least Square (WLS) approach, the state vector ‘x’ is estimated by recursively forming the Jacobian matrix H(x)= ∂ h(x) / ∂x, and solving the gain matrix equations: t
G.∆x = H .W .( z − h ( x ))
(2)
t
with G = H .W .H , where “W” is the diagonal matrix representing the inverse of the covariance matrix of “w”. 3 Review of the Observability Theory Based on the Triangular Factorization and Path Graph Concepts Using the linearized state estimator model, and calling
1996; Monticelli and Wu, 1985a; Monticelli and Wu, 1985b] will be reviewed. Property 1: If the system is observable, the factoriDC
zation of Gθ
, when a phase angle of the network
is not defined as reference, results in only one connected path graph [Bretas, 1996] and reduces that matrix to the following form [Bretas, 1996; Monticelli and Wu, 1985a; Monticelli and Wu, 1985b]: 1........................ nb 1 DC
Uθ
=
(4) (1)
0
nb
Being “nb” the number of buses (the shaded area corresponds to possible non-zero elements). Property 2: If during the factorization of Gθ
DC
one
Zero Pivot (ZP) appears in the diagonal (i,i), being “i < nb”, the system is not wholly observable and the remaining row and column entries corresponding to that position are zeros [Monticelli and Wu, 1985a; Monticelli and Wu, 1985b]. This means the remaining nodes, corresponding to the columns of U θ
DC
,
from “i+1” to “nb”, will be part of other(s) path graph(s) not having path connection(s) with the previous ones [Bretas, 1996]. The factorized form of Gθ
DC
will be [Bretas, 1996; Monticelli and Wu,
1985a; Monticelli and Wu, 1985b]: i
GθDC the resulting gain matrix, the state estima-
tor will compute the voltage phase angles by solving the equations:
GθDC θ = H PTθ WP Z P DC
with Gθ
=
T H Pθ W P H Pθ
(3)
tem Gθ
θ =0
i
(3)
0
0
(5)
, where: HPθ is the Jaco-
bian matrix related to active power measurements; ZP is the measurements vector of active power; WP the matrix of weighing factors (related to active power measurements); For observability purposes, what really matters is if the system given by equation (3) does have solution. As the number of measurements is greater than the number of states to be estimated, it is a common practice to attribute zero values for the right side of that equation, that is, one needs to know if the sysDC
UθDC =
does have solution [Bretas, 1996;
Monticelli and Wu, 1985a; Monticelli and Wu, 1985b]. So, observability analysis can be made DC
through the triangular factorization of Gθ
. In the
following, some properties demonstrated in [Bretas,
0 Remark 1: From the Property 2 one can affirm that the number of zero pivots encountered in the factoriDC
zation of Gθ
is equal to the number of path graphs
associated with this factorization Property 3: Observable islands identification: if in DC
the triangular factorization of Gθ
more than one
path graph happens, and: i) If in the available measurement set there is no injection measurement relating nodes of different path graphs, then the system as a block is unobservable and each subnetwork associated to each isolate path graph constitutes one observable island of the net-
work [Bretas, 1996]; ii) If in the available measurement set there are injection measurements relating nodes of different path graphs, then the system as a block is unobservable and it is not possible to assure the subnetworks associate with each isolated path graph constitute observable islands. In order to find observable islands, identify those measurements and discard them (these are irrelevant measurements regarding state estimation to the observable islands). Re-factorize the new matrix. Repeat the process of discarding and refactorization up to there is no injection measurement relating nodes of different path graphs. The resulting path graphs will indicate isolate observable islands [Bretas, 1996]. 4 Proposed Approach The proposed approach comprises three phases: Phase 1: Determination of the measurement set that will be used to the parameter/state estimation. That is, the measurement set which vary over a small range in a certain time period; Phase 2: Observability analysis to the proposed state/parameter estimator; Phase 3: Parameter/state estimation process. Each one of these phases will be presented in the following. 4.1 Determination of the measurement set that will be used (Phase 1) In order to determine the measurement set that will be used to the parameter/state estimation, the first step is to determine the states that vary over a small range in a certain time period. Determining these states, the measurements obtained in all the snapshots, within that time period, that are incident only to these states, will be selected to be used in the parameter/state estimation process. One state is classified as varying over a small range in a time period if its range in this period was less or equal to 1% (the threshold used to test the convergence of the state estimator). Remark 2: Before presenting the proposed method to observability analysis, the formulation of the proposed parameter/state estimation will be developed (Phase 3). 4.2 Proposed Parameter/State Estimator (Phase 3) The proposed estimator increases both, the state vector and the measurement vector. The state vector, here called Augmented State Vector (xAug), is increased with the system parameters to be estimated. Assuming that a subset of the measurements varies over a small range in a certain time period, the meas-
urement vector, here called Augmented Measurement Vector (zAug), is increased to include the measurements obtained in all snapshots within that time period. Considering the proposed augmented model, equation (1) becomes:
z Aug = h Aug ( x Aug ) + w Aug
(6) The Augmented State Vector “xAug” is obtained by recursively forming the Augmented Jacobian Ma∂h trix H Aug = Aug , and solving the Aug∂x Aug mented Gain Equation: G Aug .∆x Aug = [ H
t
Aug
] .[W
Aug
].[ z Aug − h ( x
Aug
)]
(7) t
With G Aug = [ H ] .[W ].[ H ] Aug Aug Aug The parameters to be estimated by the proposed method are the series conductances (Gkm) and susceptances (Bkm) of the transmission lines as well as their shunt susceptances (Bkmsh). Consequently, a system with “L” branches and “nb” buses has N = 2nb-1+3L” Augmented States to be estimated (being “nb” voltage magnitude measurements, “nb1” voltage angles and “3L” parameters).
4.3 Proposed Method to Observability Analysis (Phase 2) As the proposed state/parameter estimator requires the factorization of GAug, the proposed method to observability analysis is based on the factorization of that matrix and concepts contained in factorization paths. In reality the proposed method is an extension of the method proposed by Bretas [Bretas, 1996], that is based on the Gain matrix of the nonaugmented model (some of its properties are posed in section 3). Considering the formulation presented in the previous subsection, HAug has the following structure:
H Aug
H Pθ = H Qθ H Vθ
H Pv H Qv H Vv
H Pp ∂∂Pθ H Qp = ∂∂Qθ H Vp ∂∂Vθ
∂P ∂v ∂Q ∂v ∂V ∂v
∂P ∂p ∂Q ∂p ∂V ∂p
(8) where: - “P”, “Q” and “V” indicate the measurements vector of active power, reactive power and voltage measurements respectively; - “θ”, “v” and “p” indicates the voltage angle, voltage magnitude and parameter vectors to be estimated respectively. The associated matrix GAug is: Gθ Gθv Gθp t −1 G Aug = [ H Aug ] [W Aug ] [ H Aug ] = Gvθ Gv Gvp G pθ G pv G p (9) Considering this matrix and the theory presented in section 3, one can affirm that if the system is observ-
able, considering the proposed augmented model, the triangular factorization of GAug, when a phase angle of the network as reference is not defined, results in only one ZP and reduces that matrix to the following form: nb
we are considering that the measurements to be used have already been selected.
5.1 Example 1 The proposed approach is applied to the 3-bus system shown in Fig. 1. 1 2 3
UAug = nb
0
0
(10) P12,Q12 V1
P21,Q21
P23,Q23
P32,Q32
V2
Where:
V3
Î Flow measurement Î Voltage measurement If the system is not wholly observable, the factorization of GAug results in more than one ZP. In this situation it is necessary to identify the observable islands. Basing on the theory presented in [Bretas, 1996], whose propositions were presented in section 3, the identification of the observable islands is made through the following steps: Step 1: Identify the non-observable parameters. These parameters correspond to the ZPs that appear in the factorization of GAug from diagonal “2nb” up to “N + 1” (GAug is a [(N+1) x (N+1)] matrix); Step 2: If there is no power measurement relating the non-observable parameters, identify the path graphs associated with the factored submatrix Uθ of UAug (each isolate path graph constitutes one observable island); stop. Otherwise, go to the next step; Remark 3: If there is no power measurement relating the non-observable parameters, as these parameters are incident to the branches of the system, one can affirm that there is no power measurement relating nodes of different path graphs. As a consequence, each path graph constitutes, by Proposition 3, one observable island. Step 3: Remove from the selected measurements those power measurements relating the nonobservable parameters (these are discardable measurements regarding parameter/state estimation to the observable islands). Update the factorization of GAug and return to step 1. Once determined the observable islands, the parameter/state estimation will be made separately to each one of these islands. Remark 4: The observability analysis is made through the GAug matrix obtained in the first iteration.
Figure 1 Phase 1: As previously mentioned, let’s consider that the measurement set indicated in Fig. 1 were selected in order to be used. Phase 2: Observability analysis: the corresponding GAug matrix is obtained and its factorized form is: θ1 θ2 θ3
U Aug =
Three examples to characterize the proposed approach are presented in the following. In all of them
V1
V2
−0.2421 −0.1989
G23
B12
sh
B23
B12
sh
V3
G12
0
0.2078
0
0.3013
0
0.3013
B23
0
0
0.1468
0
0.1795
0
0.1795
0 0.1384
0 0
0 −0.0965
0 0
0 −0.2121
0 0
0.0366 0.0155
− 0.0331 0.0299
0.1384 0.0584
−0.071 0.0929
0.1185
− 0.6233 −0.0973 − 0.8580
− 0.1772
1 0
− 0.0252 −0.7979 −1.1013 1 0.0008 −8.0369
1 −1
0
0
1
−1
0
− 0.1327 −0.1284
0 0
0 0
0 0
0 1
0 − 0.4685
0 0
0 0
0 0
0 0
1 0
0
0
0
0
0
0
1
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
1 0
0.7882 1
− 20.7703 0.0059
0
0
0
0
0
0
0
0
0
0
0
1
0 0
− 0.2787 − 0.0554 0.0404 − 0.0234 −0.0368 1
−1.1696 0.0113
As only one ZP appears, the element UAug(θ3,θ3), the system is observable. Phase 3: Parameter/state estimation: Considering the measurement set indicated in Fig.1 and using only exact measurements, obtained through an AC power flow, the Augmented State Vector is estimated. The measurement values for that estimation are: P12
z = [0.3135
P21 P23 P32 Q12 Q21 Q23 Q32 V1 V2 V3 T − 0.2938 − 0.1055 0.1079 − 0.0183 0.0771 − 0.0056 − 0.0222 1 0.9866 1
]
Table I shows the results obtained in this phase. Remark 5: Wrong parameters were used as initial conditions. Table I – Test 1 results θ1 θ2 θ3 V1 V2 V3 G12 G23 B12 B23
5 Example
θ1 θ2 θ3 V1 V2 V3 G12 G23 B12 B23 sh B12 sh B23
B12sh B23sh
Initial conditions
Estimated values
True values
0 0.2 0.3 1 0.8 0.7 0.18 0.17 -0.8 -0.7 0.01 0.015
0 -0.3238 -0.2049 1 0.9866 1 0.1918 0.1701 -0.9591 -0.8925 0.0201 0.0205
0 -0.323 -0.213 1 0.9866 1 0.1923 0.1923 -0.9615 -0.9615 0.02 0.02
5.2 Example 2
P12
P21
z = [0.3135
The proposed method is again applied to the 3-bus system shown in Fig. 1. But now the measurement set selected to be used is: P12, P21, P23, Q12, Q21, Q23, V1, V2 and V3 (without measurements P32 and Q32). Phase 2: Observability analysis: The associated GAug matrix is obtained and the factorization results in: θ1 θ2 θ3 θ1 θ2 θ3 V1 V2 U Aug = V3 G12 G23 B12 B23 sh B12 sh B23
V1
V2
V3
1 −1 0 −0.2421 −0.1989 0 1 −1 0 − 0.2655 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
1 0 0 0 0 0 0
0 0
0 0
0 0
0 0
0 0 0
G12
G23
B12
0.2078 0 0
0 0.1468 0
0.3013 0 0
B23
B12
B23
0 0.1988 0
0.3013 0 0
0 0.5265 0
0 0.1384 0 0 − 0.4685 −0.0965 1 0.0420 −0.0208 −0.2010 −0.0635 0.0208 0 1 0.0213 −0.0180 −0.0287 0.0119 0 0 1 0.0891 −0.6225 −0.0662 0 0 0 1 −0.0254 −0.7424 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0
0 0
0 0
0 0
sh
0 0
0 − 0.2121 0.1586 −0.2106 0.0450 0.2164 − 0.8237 −0.6708 −1.1102 −7.5266 0 −8.0379 0 0 1 0
0 0
As there are three ZPs, elements UAug(θ3,θ3), UAug(B23,B23) and UAug(B23sh,B23sh), the system is not wholly observable. Then it is necessary to identity the observable islands. Step 1: The non-observable parameters are: B23 and B23sh; Step 2: The power Measurements P23 and Q23 relate the non-observable parameters; Step 3: P23 and Q23 are removed. The new GAug is obtained and factored: θ1 θ2 θ3
θ1
V1
V2
V3
0
0
sh
G23
0 0
0.3013 0
B12
B23
0 0
0.3013 0
B12
B23
0
0
0
0
0
0
θ2 θ3
0
0
0
0
V1
0 0
0 0
0 0
1 0
0 0
0 0
0 0
0 0
0 0
1 0
0 1
0 0 0 0 0 0 −0.6202 0 −0.7247 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 1
0 0 0 0 −8.0379 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 1
0 0
0
0
0
0
0
0
0
0
0
0
0
0
V2
U Aug =V3
G12 G23 B12 B23 sh
B12
sh
B23
V1 V2 T
]
Table II – Test 2 results Initial conditions
Estimated values
True values
0 0.2 1 0.8 0.18 -0.8 0.01
0 -0.3238 1 0.9866 0.1918 -0.9591 0.0201
0 -0.332 1 0.9866 0.1923 -0.9615 0.02
θ1 θ2
V2 G12 B12 B12sh
5.3 Example 3 Let’s consider the 3-bus system shown in Fig.1 again. But now the selected measurements are: P12, Q12, P23, Q23, P32, Q32, V1, V2 and V3. Phase 2: Observability analysis: The factorized GAug matrix is: θ1 θ2 θ3 V1 V2 U Aug = V3 G12 G23 B12 B23 sh B12 sh B23
θ1 θ2 θ3
V1
V2
V3
1 −1 0 − 0.4842 0 0 0 1 −1 0 − 0.1327 − 0.1284
G12
0.2013 0
0
G23
B12
0 0.1468
0
0
sh
B23
B23
0 0.1795
0.6234 0
0 0.1795
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
1 0
− 0.409 1
0
0
0
0
0
1
0
0
0
0
0
0
1
0.1349
0 0
0 0
0 0
0 0
0 0
0 0
0 0
1 0
0 0
− 0.7976 0
0
0
0
0
0
0
0
0
0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0.1164 −0.2833 − 0.0511
sh
B12
0.3557 0
− 0.0659 0 − 0.4552 0 0.0289 − 0.0337 0.1996 − 0.0721
0 0.0411
− 0.0216 − 0.0366
0.0122
0.0298
0.0845
0.0926
− 0.5659 − 0.1109 −3.9106 − 0.2018 0 0
−1.1662 0
1
0
− 20.775
0 0
1 0
0 1
sh
G12
1 −1 0 −0.2421 −0.1929 0 0 0 0 0 0 0
0.2078 0
Q21
Table II shows the results obtained in this phase.
V1 sh
Q12
− 0.2938 − 0.0183 0.0771 0.9866 1
0 0
−0.4685 0 0.1384 0 −0.0965 0 −0.2121 0 1 0 −0.0811 0 0.0536 0 0.2024 0
There are two ZPs, elements UAug(θ3,θ3) and UAug(B12,B12). Removing the measurements P12 and Q12, that relating the non-observable parameter B12, the new factorized form of GAug is: θ1 θ2 θ3 V1 θ1 θ2 θ3 V1 V2 V3
There are five ZPs: UAug(θ2,θ2), UAug(θ3,θ3), UAug(B23,B23) and UUAug(G23,G23), sh sh Aug(B23 ,B23 ). Return to step 1. Step 1: The non-observable parameters are: G23, B23 and B23sh; Step 2: As there is no power measurement relating the non-observable parameters, the two path graphs associated with the factorized submatrix Uθ of UAug constitute two observable islands, being one formed by the buses 1 and 2 and other formed only by the bus 3. Remark 6: The parameter/state estimation is not processed in case the Observable Island is formed by only one isolate bus. Phase 3: Parameter/state estimation: Considering the observable island formed by buses 1 and 2, the parameter/state estimation is processed. The measurement values for that estimation are:
U Aug = G
12
G23 B12 B23 sh
B12
sh
B23
V2
V3
G12
0 0 0 0 0 0 0 0 1 −1 0 −0.1327 −0.1284 0 0 0 0 0 −0.3453 0
G23
0 0.1468 0 0 0.05
B12 0 0
B23
0 0.1795
sh
B12 0 0
sh
B23
0 0.1795
0 0 0 0 0 0
0 0 0
0 0 0
0 0 1
0 0 0 0 0 0 0 0 0 −0.0410 0 −0.0879
0 0 0 0
0 0
0 0
0 0
1 0
0 −0.0337 0 0 0 0
0 0 0 0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
1 0 0
0 −0.7976 0 −1.1662 0 0 0 0 0 1 0 − 20.775
0 0 0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0.0274 0
0 0
0 0
0.0882 0
0 0
0 1
There are six ZPs, elements UAug(θ1,θ1), UAug(θ3,θ3), UAug(V1,V1), UAug(G12,G12), UAug(B12,B12) and UAug(B12sh,B12sh). Using path graph concepts one finds two observable islands, one formed by bus 1 and other formed by buses 2 and 3. Phase 3: Parameter/state estimation: The parameter/state estimator is processed to the island formed by buses 2 and 3. The measurement values for that estimation are: P23
z = [− 0.1055
P32
Q23
Q32
V2
V3
]T
0.1079 − 0.0056 − 0.0222 0.9866 1
Table III shows the estimation results.
Table III – Test 3 results θ2 θ3 V2 V3 G23 B23 B23h
Initial condition
Estimated Values
Real Values
0.2 0.3 0.8 0.7 0.17 -0.7 0.015
0.2 0.3189 0.9866 1 0.1701 -0.8925 0.0205
-0.323 -0.213 0.9866 1 0.1923 -0.9615 0.02
Remark 7: Considering the same system, with the wrong parameters, and the same exact measurement sets used in those test, one traditional WLS state estimator was processed. In all the tests, besides to allow the parameter estimation, the voltages and phase angles estimated by the parameter/state estimator were closer to the true vales than those estimated by the traditional WLS state estimator. 6 Conclusions In this paper, an approach to parameter and state estimation based on the WLS normal equations was proposed. This approach augments both the system state vector and the measurement vector. The state vector to be estimated includes some branch parameters together with the bus voltage and phase angles while the measurement vector includes all those measurements that remain nearly unchanged over a period of time. In order to determine the portion of the system whose parameters and states can be estimated through the selected measurements, an observability analysis method was proposed. This method is based on the factorization of GAug and makes use of the concept of factorization paths. The proposed approach is still in a study phase and this paper presented only the initial results, which are very promising. 7 Acknowledgment The authors would like to acknowledge the FAPESP for the financial support. 8
References
Abur, A. and Expósito, A.G. (2004). Power system state estimation: theory and implementation. Marcel & Dekker Publishers, Nova York, EUA Alsaç, O., Vempati, N. Sttot, B. and Monticelli, A. (1998). Generalized state estimation, IEEE Transactions on Power Systems, Vol. 13, No. 3, pp.1069-1075, Aug. Bretas, N.G. (1996). Network Observability: Theory and Algorithms based on Triangular Factorisation and Path Graph Concepts, IEE Pro-
ceedings: Generation, Transmission, and Distribution, Vol. 143, No. 1, pp.123-128, Jan. Debs, A. (1974). Estimation of steady-state power system model parameters. IEEE Transactions on Power Apparatus and Systems, Vol.PAS-93, No. 5, pp.1260-1268. Monticelli, A. J. and Wu, F. F. (1985a). Networ Observability: Theory”, IEEE Transactions on Power Systems, Vol. 104, No. 5, pp.1042-1048, May. Monticelli, A. J. and Wu, F. F. (1985b). "Network Observability: Identification of Observable Islands and Measurement Placement”, IEEE Transactions on Power Systems, Vol. 104, No. 5, pp.1035-1041, May. Liu, W. and Lun, S. (1992). Estimation of parameter errors from measurement residuals in state estimation, IEEE Transactions on Power Systems, Vol. 7, No. 1, pp.81-89, Feb. Liu, W. and Lun, S. (1995). Parameter error identification and estimation in power system state estimation, IEEE Transactions on Power Systems, Vol. 10, No. 1, pp.200-209, Feb. Slutsker, I. and Clements, K. (1996). Real time recursive parameter estimation in energy management systems. IEEE Transactions on Power Systems, Vol. 11, No. 3, pp.1393-1399, Aug. Van Cutsen, T. and Quintana, V.(1988). Network parameter estimation using online data with application to transformer tap position estimation, in IEEE Proceedings, Vol. 135, pp.31-40, Jan. Zarco, P. and Expósito, A.G. (2000). Power system parameter estimation, IEEE Transactions on Power Systems, Vol. 15, No. 1, pp.216-222, Feb.
